On the number of matroids

TL;DR

This paper tightens bounds on the number of matroids on n elements, showing that the logarithm of the logarithm of their count is close to Knuth's lower bound, using structural and combinatorial properties.

Contribution

It proves an upper bound on the number of matroids that closely matches Knuth's lower bound, refining the asymptotic estimate.

Findings

Established an upper bound within an additive term of Knuth's lower bound.

Used structural properties of non-bases and Johnson graph to represent matroids efficiently.

Confirmed that the number of matroids grows roughly as suggested by earlier bounds.

Abstract

We consider the problem of determining $m_n$, the number of matroids on $n$ elements. The best known lower bound on $m_n$ is due to Knuth (1974) who showed that $\log \log m_n$ is at least $n-3/2\log n-1$. On the other hand, Piff (1973) showed that $\log\log m_n\leq n-\log n+\log\log n +O(1)$, and it has been conjectured since that the right answer is perhaps closer to Knuth's bound. We show that this is indeed the case, and prove an upper bound on $\log\log m_n$ that is within an additive $1+o(1)$ term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of independent sets in the Johnson graph to give a compressed representation of matroids.